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The answer is "no":

Take $M>2$ and evaluate

$$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}),$$ so $z_1$ and $z_2$ are correlated no matter how large $M$.

The answer is "no":

Take $M>2$ and evaluate

$$a\,\mathbb{E}[|z_1|^2]=a\,\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$a^2\,\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow a^2\,\mathbb{E}[|z_1|^2|z_2|^2]-a^2\,\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}),$$ so $z_1$ and $z_2$ are correlated no matter how large $M$.

The answer is "no":

Take $M>2$ and evaluate

$$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}),$$ so $z_1$ and $z_2$ are not independent, no matter how large $M$.

The answer is "no":

Take $M>2$ and evaluate

$$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}),$$ so $z_1$ and $z_2$ are correlated no matter how large $M$.

The answer is "no":

Take $M=6>5$ and evaluate $$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{60},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{3360},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{50400}\neq 0,$$ so $z_1$ and $z_2$ are not independent.

More generally, for any $M>2$, one has $$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}).$$

The answer is "no":

Take $M>2$ and evaluate

$$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}),$$ so $z_1$ and $z_2$ are not independent, no matter how large $M$.